This is currently a grey area. The uncertainty began with the 2013 Alice ruling, and is still being worked out.
After Alice, at a high-level, the knee-jerk reaction was that game mechanics and software would no longer be patent eligible. Since that ruling, which is problematic in that what constitutes "abstract" is not well defined, there has been some significant walk-back.
Currently there is a trend to tie software and game mechanics into databases because of a software grant involving databases. (This is merely a strategy for dealing with the review board, and doesn't get to the root of the issue.)
I can get more technical, and provide some examples, if requested, but I've covered this subject in depth in other answers.
The best legal advice I ever got in this regard is that the only certainty in patent law is uncertainty (continual chaos) regarding the guidelines.
The US has a long history of precedent regarding the eligibility of novel game mechanics, and there seem to have been post-Alice grants that constitute pure methods.
Speaking from experience, a major consideration in whether to pursue a game is cost/benefit. Utility patent applications can be expensive with no guarantee of a grant. The current "grey area" status for game mechanics significantly increases that risk.